Purdue University Graduate School
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Duality and Local Cohomology in Hodge Theory

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thesis
posted on 2023-04-25, 22:41 authored by Scott M HiattScott M Hiatt

A Hodge module on an algebraic variety may be viewed as a variation of Hodge structure  with singularities. Given an irreducible variety $X$, for any polarized variation of Hodge structure $\bold{H}$ on a smooth open subvariety $U\subset X,$ there exists a unique Hodge module $\cM \in HM_{X}(X)$ that extends $\bH.$ Conversely, for any Hodge module $\cM \in HM_{X}(X)$ with strict support on $X,$ there exists a polarized variation of Hodge structure $\bH$ on a smooth open subset $U \subset X$ such that $\cM \vert _{V} \cong \bH.$ In this thesis, we first study the singularities of a Hodge module $\cM \in HM_{X}(X)$ by using Morihiko Saito's theory of $S$-sheaves and duality. Then using local cohomology and the theory of mixed Hodge modules, we study the Hodge structure of $H^{i}(X, DR(\cM))$  when $X$ is a projective variety. Finally, we consider a variation of Hodge structure $\bH$ on $U$ as a Hodge module $\cN \in HM(U)$ on $U,$ and study the local cohomology of the complex $Gr^{F}_{p}DR(j_{!}\cN) \in D^{b}_{coh}(\cO_{X}),$ where $j: U \hookrightarrow X$ is the natural map.

History

Degree Type

  • Doctor of Philosophy

Department

  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Donu Arapura

Additional Committee Member 2

Kenji Matsuki

Additional Committee Member 3

Jaroslaw Wlodarczyk

Additional Committee Member 4

Uli Walther

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